Saturday, 19 May 2012

There is a well-known paradox of the surprise examination. A teacher tells some students that there will be an examination one morning next week (Monday to Friday), but that they will not know until the morning of the examination that it is on that day.

The students reason that it cannot be on Friday, because then they could work out the day by Thursday evening. But if they know it cannot be on Friday, it cannot be on Thursday either, because they would be able to work out the day by Wednesday evening. They continue to reason on these lines, and conclude that the examination cannot be held at all, under the conditions stated by the teacher.

It is important that the timespan over which the students would be at risk has an end-point, Friday, backwards from which the students can reason. If the days at risk went on for ever, they could not reason as they do. (It is an interesting question, whether the end-point needs to be a determinate one, up to which they are at risk. I think that is not necessary. That is, it would suffice for them to be able to identify a specific date, a finite time in the future, beyond which the period of risk had definitely ended. It would not matter whether they were definitely at risk up to that date.)

It is also important that the examination would definitely be held within the finite timespan. If the teacher only said that there might be an examination next week, and that if there were, it would be held on a day that the students could not predict before that day, the students could come to school each day, unsure of whether there would be an examination that day.

It is also important that the students would definitely be unable to identify the day of the examination before that day. If they were told only that they might not be able to identify the day in advance, then Friday would be a possible day. Then it could not be eliminated, and the other days would also be possible days.

We may note that there is no requirement for the teacher to have decided, at the time of the announcement to the students, the day of the examination. The paradox would arise if the teacher planned to pick a day on impulse, as the week wore on. The teacher, wanting to meet the conditions, would know that he or she could not leave the examination until Friday, and would also know that the students knew this. The teacher would therefore know that he or she could not leave the examination until Thursday, and would also know that the students knew this. This chain of reasoning in the teacher's mind would lead the teacher to the same conclusion as the students. The examination could not be held at all, under the stated conditions, even though the teacher's initial proposal of a surprise examination looked perfectly reasonable.

We can also see that human intention only matters to make the story plausible, not to generate the paradox once we have the story, by considering a predicted earthquake.

Having studied the ways in which stresses have built up, we may conclude that there will be an earthquake at a given location on some day between now and the end of 31 December 2100, that it will be the only earthquake there before 2500 (because stresses will be relieved by the earthquake and will take a while to build up again), and that we will not know, until the day of the earthquake, that it is on that day.

Just to make the problem as similar as possible as that of the examination, let us assume that our methods give us no way of telling when the probability of an earthquake within the next year, or the next month, or any other future time period shorter than the time to 31 December 2100, is rising (apart from the rise due to the fact that as time passes, there are fewer days left to go). We cannot see it coming, and each remaining day in the given time period will always seem to us as likely as any other remaining day.

The lack of human agency makes the story implausible. There is no-one out to keep us in ignorance. But if we can overlook that, we can see that given the conditions, the earthquake could not be on the last day, because we would know by the end of 30 December 2100 that it would be on that day. Therefore it could not be on the penultimate day, and the paradox would arise all over again.

One approach would be to say that while the students, or the potential victims of the earthquake, could not say that they were in some unidentified member or other of a set of possible worlds (the examination-set being the world in which the examination is on Monday, the world in which it is on Tuesday, and so on to Friday), they could say that they were in some superposition of worlds, and that this superposition would be collapsed into a particular world by the teacher's announcement, or by the first movement of the tectonic plates. Analogies with quantum mechanics may be fun, but I fear that they can also degenerate into mere hand-waving. I shall not pursue this approach here.

Instead, we can improve our understanding by looking at a feature of the mathematical structure of the problem. Not only is there a finite timespan of risk. That timespan contains a finite number of risk-points, the individual days. For each such risk-point, there is an immediately preceding risk-point, the day before. That allows the paradox to be generated. To take the example of the examination, there is a last risk-point, Friday. The examination cannot be held on that day. Therefore, the immediately preceding risk-point, Thursday, effectively becomes the last risk-point. But the examination cannot be held at the last risk-point, so Thursday is eliminated, Wednesday effectively becomes the last risk-point, and the reasoning is repeated.

Suppose instead that the teacher had said that an examination of two hours would start at some unexpected moment within school hours, say 0900 to 1700, on some day within the next week. The examination would have to start by 1500, but there would still be an infinite number of moments in each day at which it could start. (We shall assume that time is infinitely divisible.) Then the students' reasoning would be blocked.

It would be blocked because moments of time are densely ordered. That is, between any two moments in order of time, there is another moment. Between 0901 and 0902, there is 0901 + 30 seconds. Between 0901 and 0901 + 30 seconds, there is 0901 + 15 seconds. However finely we chop up time, even into microseconds, there will always be more moments in between the ones that we have already identified. And each moment would be a risk-point, if the teacher had announced that the examination would start at some unexpected moment.

The dense ordering would be enough to block the students' reasoning. Their reasoning relied on deleting the last risk-point, and moving back to the immediately preceding risk-point. But if the risk-points were densely ordered, there would be no immediately preceding risk-point, to which they could move. Any preceding risk-point they identified would not be the immediate predecessor of the point they had deleted, because there would be another risk-point, later than that one but earlier than the point they had deleted.

Thus a densely ordered stretch of risk-points blocks the chain of reasoning. It is not even necessary for the whole stretch of risk-points to be densely ordered. (Indeed, it is not densely ordered in the case of the examination. There is, for example, no risk-point between the one at 1500 on Monday and the one at 0900 on Tuesday.) All that is necessary is for there to be some densely ordered stretch of risk-points before the last risk-point. If there is a stretch reaching back from that last point that is not densely ordered, the reasoning of the paradox can be used to eliminate all of the risk-points in that stretch, but once we have eliminated the risk-point at the end of the last densely ordered stretch, the reasoning is blocked.

(The last densely ordered stretch might be open at its end, that is, it might not include its end point. It might, for example, be all moments from 0900 on Thursday up to, but not including, 1200 on Thursday. In that case, there would be no final risk-point of the stretch to be eliminated. One could only work back as far as eliminating the risk-point that came first in order of time after the last densely ordered stretch.)

Another feature of the arrangement, which would arise outside densely ordered stretches of risk-points, but which would not arise within any densely ordered stretch, is that there would be an interval between each risk-point (other than the final one) and a risk-point that came later, which was itself empty of risk-points, and within which the students could come to appreciate that only the later risk-points were still possibilities. This matters because such appreciation would be the mechanism, by virtue of which it would be impossible for the students to have the promised ignorance of which risk-point was the one at which the examination would occur. In the interval between the penultimate and the final risk-points, satisfaction of the condition of ignorance would rule out the final risk-point.

My thoughts turned to the paradox because of Greece's likely exit from the Euro. It is essential for currency reforms to be kept secret until they take effect. Otherwise the withdrawal of funds, and speculation, will lead to a bigger crisis than the one that the reform is intended to resolve. Witness, for example, the highly secretive preparations for the introduction of the Deutsche Mark in 1948, first the Konklave von Rothwesten, at which the reform was worked out, and then the delivery of banknotes, in advance of the announcement, in Operation Bird Dog. Moreover, currency reforms take place on specific days, not at specific moments (except, possibly, midnight at the start of some day.) So is there scope for a paradox of the surprise currency reform?

There is one fact that prevents such a paradox from arising, and it is a fact that politicians happily exploit, although perhaps not out of a conscious desire to avoid the trap of this paradox. The relevant fact is that there is no known end-point to the period of risk. Even if one is certain that Greece will leave the Euro, and however bad things get, there is always the possibility that some new fix will tide Greece over for a little bit longer.

Alternatively, people might become certain that the fixes would run out, and that Greece would leave the Euro within a period with an end-point that was already known. But once people became certain of that, the markets would not sit around like students, worrying whether the examination would be today, tomorrow or the next day. They would respond immediately, as if the examination had just started, with only a very limited softening of their reaction in recognition of the fact that the exit might still be delayed by a few months.